Let $$\vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k}$$ and $$\vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k}$$. Let $$\vec{\beta_1}$$ be parallel to $$\vec{\alpha}$$ and $$\vec{\beta_2}$$ be perpendicular to $$\vec{\alpha}$$. If $$\vec{\beta} = \vec{\beta_1} + \vec{\beta_2}$$, then the value of $$5\vec{\beta_2} \cdot (\hat{i} + \hat{j} + \hat{k})$$ is

We are given $$\vec{\alpha}=4\hat{i}+3\hat{j}+5\hat{k}$$ and $$\vec{\beta}=1\hat{i}+2\hat{j}-4\hat{k}$$, and we decompose $$\vec{\beta}$$ into a component parallel to $$\vec{\alpha}$$, denoted by $$\vec{\beta_1}$$, and a component perpendicular to $$\vec{\alpha}$$, denoted by $$\vec{\beta_2}$$, so that $$\vec{\beta}=\vec{\beta_1}+\vec{\beta_2}$$. Our goal is to evaluate $$5\vec{\beta_2}\cdot(\hat{i}+\hat{j}+\hat{k})$$.

By the definition of vector projection, we have

$$\vec{\beta_1}=\frac{\vec{\beta}\cdot\vec{\alpha}}{|\vec{\alpha}|^2}\,\vec{\alpha}$$

We first compute the scalar product

$$\vec{\beta}\cdot\vec{\alpha}=(1)(4)+(2)(3)+(-4)(5)=4+6-20=-10$$

and the squared magnitude of $$\vec{\alpha}$$

$$|\vec{\alpha}|^2=4^2+3^2+5^2=16+9+25=50$$

Substituting these values into the formula for $$\vec{\beta_1}$$ gives

$$\vec{\beta_1}=\frac{-10}{50}\,\vec{\alpha}=-\frac{1}{5}(4\hat{i}+3\hat{j}+5\hat{k})=-\frac{4}{5}\hat{i}-\frac{3}{5}\hat{j}-\hat{k}$$

The perpendicular component then follows from the relation $$\vec{\beta_2}=\vec{\beta}-\vec{\beta_1}$$, which yields

$$\vec{\beta_2}=\left(1+\frac{4}{5}\right)\hat{i}+\left(2+\frac{3}{5}\right)\hat{j}+(-4+1)\hat{k}=\frac{9}{5}\hat{i}+\frac{13}{5}\hat{j}-3\hat{k}$$

Multiplying $$\vec{\beta_2}$$ by 5 gives

$$5\vec{\beta_2}=9\hat{i}+13\hat{j}-15\hat{k}$$

Finally, taking the dot product with $$\hat{i}+\hat{j}+\hat{k}$$ leads to

$$5\vec{\beta_2}\cdot(\hat{i}+\hat{j}+\hat{k})=9+13-15=7$$

Hence the required value is 7.